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Interpolated Sequences and Critical L-Values of Modular Forms

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Book cover Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

Recently, Zagier expressed an interpolated version of the Apéry numbers for \(\zeta (3)\) in terms of a critical L-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier’s six sporadic sequences are essentially critical L-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown’s cellular integrals and critical L-values of modular forms of odd weight.

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Acknowledgements

The first author would like to thank the Hausdorff Research Institute for Mathematics in Bonn, Germany for their support as this work began during his stay from January 2–19, 2018 as part of the Trimester Program “Periods in Number Theory, Algebraic Geometry and Physics”. He also thanks Masha Vlasenko for her support and encouragement during the initial stages of this project. The authors are particularly grateful to Wadim Zudilin for sharing his proof of Theorem 2 for sequence \(\varvec{F}\).

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Authors and Affiliations

  1. School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

    Robert Osburn

  2. Dept. of Mathematics and Statistics, University of South Alabama, 411 University Blvd N, Mobile, AL, 36688, United States

    Armin Straub

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  1. Robert Osburn
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Correspondence to Armin Straub .

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Editors and Affiliations

  1. Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

    Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

  3. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Peter Paule

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Osburn, R., Straub, A. (2019). Interpolated Sequences and Critical L-Values of Modular Forms. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_14

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  • DOI: https://doi.org/10.1007/978-3-030-04480-0_14

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